Families of L-functions and Applications

研究者将继续研究L-函数。 特别是一般自守L-函数的“次凸”估计问题。 这些有广泛的应用经典问题在数论（例如希尔伯特的第11个问题表示整数在数域的二次形式）和本征函数的算术表面。 L-函数族和与它们相关的对称性的使用是至关重要的。这个项目是关于数学对象的研究称为zeta函数。 它们编码了关于算术和几何的深层信息，这些信息存在于它们的定义中。 理解这些函数对整数（和素数）理论有着深远的应用，更令人惊讶的是对量子理论中的一些基础问题有着深远的应用。

The investigator will continue his study of L-functions. Specifically the problem of a 'subconvex' estimate for general automorphic L-functions. These have wide applications to classical problems in number theory (e.g. Hilbert's 11-th problem on representing integers in a number field by a quadratic form) and to eigenfunctions on arithmetic surfaces. The use of the families of L-functions and symmetries associated with them is crucial. This project is concerned with the study of mathematical objects called zeta functions. They encode deep information about the arithmetic and geometry which lie at their definition. Understanding the functions has far reaching applications to the theory of whole numbers (and prime numbers) and more surprisingly to some foundational problems in quantum theory.